2003 Audi A4 winning noise when engine is cold

That is your Secondary Air Injection pump. It is part of the Emissions system and is normal for it to run on start up.
As long as the MIL is not on in the dash it is operating properly.

Thank you Dubtech ... I purchased this car new in 2003 and until recently I could hear it slightly ... which must be normal. However, it is now much louder than it used to be. I can hear it when my wife drives away for about 300 feet. After about 2 minutes or so, it stops completely and we will not hear it again until we start a cold engine. Do you think something could be clogged?

It is an air pump and the pitch will change based on the restriction of airflow. Sometimes the "Kombi" valve doesn't open, which then strains the pump because it is trying to push air into a closed valve. The air pump itself sometimes warps and leaks through the seams of the assembly. Both are common problems, but the ECU can and will detect any loss of flow. Since it is part of the emission system it is tightly monitored and should turn on the MIL if there was a problem..